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In mathematics, Farkas' lemma is a solvability theorem for a finite system of linear inequalities. It was originally proven by the Hungarian mathematician Gyula Farkas . [ 1 ] Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively ...
Kőnig's theorem (set theory) Kőnig's theorem (graph theory) Lagrange's theorem (group theory) Lagrange's theorem (number theory) Liouville's theorem (complex analysis) Markov's inequality (proof of a generalization) Mean value theorem; Multivariate normal distribution (to do) Holomorphic functions are analytic; Pythagorean theorem; Quadratic ...
Manin published a proof in 1963, but Coleman (1990) found and corrected a gap in the proof. In 1973 Britton published a 282-page attempted solution of Burnside's problem. In his proof he assumed the existence of a set of parameters satisfying some inequalities, but Adian pointed out that these inequalities were inconsistent.
A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are linear inequalities. [6] The list of constraints is a system of linear inequalities.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
Linear congruence theorem (number theory, modular arithmetic) Linear speedup theorem (computational complexity theory) Linnik's theorem (number theory) Lions–Lax–Milgram theorem (partial differential equations) Liouville's theorem (complex analysis, entire functions) Liouville's theorem (conformal mappings) Liouville's theorem (Hamiltonian ...
It is an important proof technique in set theory, topology and other fields. Proofs by transfinite induction typically distinguish three cases: when n is a minimal element, i.e. there is no element smaller than n; when n has a direct predecessor, i.e. the set of elements which are smaller than n has a largest element;
This article lists mathematical identities, that is, identically true relations holding in mathematics. Bézout's identity (despite its usual name, it is not, properly speaking, an identity) Binet-cauchy identity